A parabolic problem involving $p(x)$-Laplacian, a power and a singular nonlinearity

Abstract

The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-Δ_{p(x)}u&=λu^{q(x)-1} + u^{-δ(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~Σ_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~Ω\nonumber. \end{align*} Here $Q_T=\Omega \times (0,T)$, ${\Sigma }_T=\partial \Omega \times (0,T)$, $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ ($N\ge 2$) with Lipschitz continuous boundary $\partial \Omega$, $\lambda \in (0,\infty )$, $f\in L^1(Q_T)$, $g\in L^{\infty }(\Omega )$, $u_0\in L^r(\Omega )$ with $r\ge 2$, $\delta :\overline{\Omega }\to (0,\infty )$ is continuous, and $p,q\in C(\overline{\Omega })$ with $\underset{x\in \overline{\Omega }}{\max }p(x)<N$, $q(\cdot )<p^{\ast }(\cdot )$. The article is distinguished into two cases according to the choice of $f$ with different range of parameters $p(\cdot )$, $q(\cdot )$.